Did it? They didn’t get there so did we get bigger fire at their target?
I imagine the training will consist of something like changing the comms protocol to say “runway lights are on, control. Truck 1 confirming cross runway 4D?” prior to crossing. Double check so to speak.
Ground vehicles consistently have radio conventions that just don't fit into the aviation world. It feels like a contributor to this accident, you can hear the controller's brain skip a couple gears trying to understand the goofy word order from the truck.
Pilots and controllers speak the same language in the same order; ground vehicles just kinda say stuff.
The aviation-ized version of your proposal would be something like this:
> tower truck 1 short of 4 at delta, red status lights